## _wp04103

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---

### I. Main argument and mechanism
- When labor is heterogeneous and matching of skills to jobs is below first-best, introduction of trade can induce industrial agglomeration and interregional trade.
- Mechanism:
  - Agglomeration arises because the average quality of matches improves when firms in a local market have a larger pool of workers.
  - The countervailing force is trade costs (iceberg transport costs τ, τ ≥ 1).
  - At zero trade costs, regions that can trade always specialize; with positive trade costs regions may or may not specialize depending on parameters.
- Model features:
  - Heterogeneities in task performance: ex ante identical workers differ in productivity across jobs.
  - Labor productivity depends on technology, training, conventional production factors, and match quality.
  - Firms with specialized skill requirements recruit better-matched workers in larger markets.
  - Incorporates increasing returns, differentiated goods, and transport costs from new trade theory.
- Empirical and anecdotal support (quotes preserved from source):
  - Sony UK general manager: “What keeps us here is the quality of the staff and the research and development capacity” (Financial Times, January 19, 2002).
  - Gavin Clarkson: “Having a critical mass of people who are highly skilled and resourced is what attracts business to any location or community.” (Australian Financial Review, 11 April 1999).
  - Financial Times: “The survival of struggling volume producers may or may not prove directly vital to the economy as a whole, but theirrole in providing skilled staff and components infrastructure for higher-margin niche manufacturers is hard to ignore.”

### Empirical evidence on labor pooling and matching
- Dumais, Ellison and Glaeser (2002) using LRD manufacturing data for the United States:
  - Examined Marshall’s three reasons for agglomeration: proximity to suppliers/customers, labor pooling, information spillovers.
  - Found labor pooling was by far the most important force for agglomeration at the metropolitan area level.
  - New entrants tended to locate in areas where existing firms had labor requirements similar to their own.
- Interpretation and extensions:
  - Labor pooling evidence supports match-differentiation being more important for advanced skills; prediction: agglomeration should increase with economic growth and be more prevalent in industries requiring more high-tech labor.
  - Dumais, Ellison and Glaeser (2002) find labor pooling especially strong in high technology industries (fabricated metals, industrial machinery, electronic and electrical equipment, and instruments).
  - Amiti and Cameron (2004) found labor pooling benefits significant in a developing country (Indonesia), though interfirm linkages were sometimes more important.
- Additional findings:
  - Dumais, Ellison and Glaeser (1997) find labor pooling more important in industries with more volatile employment.

### Model structure and assumptions
- Geography and sectors:
  - Two regions (home and abroad), two sectors (agriculture and manufacturing), two skill levels (skilled and unskilled).
- Mobility assumptions:
  - (a) perfect inter-sectoral and inter-regional mobility of firms;
  - (b) perfect inter-sectoral mobility of labor, but movement from agriculture to manufacturing requires a fixed “training” cost;
  - (c) no inter-regional mobility of labor.
- Production and preferences:
  - Agriculture: single good, linear technology, one worker produces one unit, price and agricultural wage normalized to 1.
  - Manufacturing: many differentiated varieties, one firm per variety, Dixit-Stiglitz preferences: Uk = vk Cxk^μ Cak^(1−μ), 0<μ<1, vk>0 (eq. (1)).
  - Manufacturing price index Px as in the source; iceberg transport costs τ, elasticity of substitution σ>1, τ ≥ 1.
- Training and occupational choice:
  - Training is a proportional utility cost: vk = 1/t if trained, vk = 1 otherwise; t > 1.

### Labor-market matching, supply and wages (key expressions preserved)
- Skill heterogeneity: workers uniformly distributed on a circle of circumference 2H; heterogeneity parameter H (H = 0 implies no heterogeneity).
- Match quality: distance d on circle; effective units from a worker at distance d is 1−d (normalized to 1 at zero distance).
- Firm spacing: with N manufacturing firms the distance between neighbors is 2H/N; worst mismatch m ≡ H/N.
- Wage posting and worker attraction:
  - di = (wi − w(1−2m)) / (wi + w) (eq. (14)).
  - Total effective labor units to firm i:
    LEsi = (Ls/2H) (wi − w + 2mw)(wi + 3w − 2mw)/(wi + w)^2 (eq. (15)).
  - Elasticity of labor supply in symmetric equilibrium (as presented in source):
    ηEsi ≡ (∂LEsi/∂wi)(wi/LEsi)|wi=w = (1−m)^2 (2−m) m > 0 (eq. (16)).
  - Symmetric-equilibrium supply per firm:
    LEsi = (Ls/N)(1−m/2) (eq. (17)).
- Occupational choice and inter-sectoral mobility condition:
  - Expected utility of a trained worker:
    Ūk = t^{−1} μ^{μ} (1−μ)^{1−μ} P_x^{−μ} P_a^{−(1−μ)} w(1−m/2) (eq. (19)).
  - Inter-sectoral mobility condition with both sectors active:
    w(1−m/2) = t > 1 (eq. (20)).
  - Feasible range for mismatch:
    m ≤ (2t−2)/(2t−1) (eq. (21)).

### Firm behavior, output, and equilibrium matching
- Mark-up and pricing (as presented):
  - p_i = w (1−m)^2 σβ/(σ−1) (eq. (22)).
- Firm output and labor demand:
  - x = α(σ−1)/β (1−m)^2 σ − (σ−1)(1−m)^2 (eq. (23)). [Preserved exactly as printed]
  - Demand for effective labor per firm:
    L_EDsi(m) = ασ / (σ − (σ−1)(1−m)^2 ) (eq. (24)).
- Supply per firm expressed in m:
  - L_ESsi(m) = (Ls/H) m(1−m/2) (eq. (25)).
- Interior equilibrium condition (transform M ≡ m(1−m/2)):
  - 2(σ−1) M^2 + M − (ασH/Ls) = 0 (eq. (27)); the single positive root with feasible m < 1 is the unique equilibrium m.
- Special case H = 0 (m = 0):
  - N = Ls/(ασ) (eq. (26)).
- Comparative statics:
  - Equilibrium m increases in H and decreases in Ls; as more skilled workers enter, more manufacturing firms enter (complementarity).

### Aggregate equilibria: symmetric and agglomeration allocations
- Mean manufacturing wage set by mobility condition:
  - Ws = t (eq. (28)); m = m(Ls) with m'(.) < 0 (eq. (29)).
- Equilibrium pricing (implicit in Ls):
  - p = Ws/(1−m/2) (1−m)^2 σβ/(σ−1) (eq. (30)).
- National income with zero profits:
  - Y = Ws Ls + Wa La, Wa = 1, La = L − Ls ⇒ Y = L + (Ws − 1) Ls (eqs. (31)–(32)).
- Profit-zero condition and interregional interaction:
  - Introduces φ ≡ (m/m^*) (μ p/p^*)^{σ−1} (eq. (36)); home equation (35) and symmetric foreign equation (37) determine allocations.
- Symmetric equilibrium (μ < 0.5, L = L^*):
  - Ls = μ/[t − (t−1) μ] L ≡ μ̃ L (eq. (38)); Ls^* = Ls, W_s^* = t, φ = 1.
  - Wage per unit: w = t/(1−m/2).
  - Price:
    p = t σ β (1−m)^2 / (1−m/2)^{(σ−1)} (eq. (39)).
  - Output per firm:
    x = μ̃ L t^2 / [σ β m H (σ−1) (1−m)^2 (1−m/2)] (eq. (40)).
- Autarky: τ → ∞ yields no trade; same factor-allocation equations apply.
- Agglomeration equilibrium (all manufacturing in home country):
  - N^* = L_s^* = 0 (foreign manufacturing absent).
  - Home manufacturing employment:
    Ls = μ(L + L^*) / [(1−μ) t + μ] = 2 μ̃ L (eq. (41)).
  - In agglomeration equilibrium the foreign country produces only agriculture; home has fraction 2 μ̃ of workers in manufacturing earning wage t.
  - Firm output per firm is twice the symmetric-equilibrium expression in (40) under agglomeration.

### Selection of equilibrium: break point and sustain point (robustness)
- Approach: perturb symmetric equilibrium by moving manufacturing workers dL_s from foreign to home; evaluate sign of dW_s/dL_s.
  - If dW_s/dL_s < 0, symmetric equilibrium is robust.
  - If dW_s/dL_s > 0, symmetric equilibrium breaks (there exists a break point τ(B) where sign changes).
- Total differentiation yields (expression reproduced exactly as in source) (42) and the sign of dW_s/dL_s is given by expression (43) as presented:
  - μ̃ L (μ) dW_s + t μ dL_s = (1 − τ^{1−σ})/(1 + τ^{1−σ}) [(t−1) dL_s + μ̃ L dW_s] − [2 τ^{1−σ} t μ̃ L / μ (1 + τ^{1−σ})^2] dφ (eq. (42)).
  - The sign of dW_s/dL_s is the same as the sign of:
    − t/μ + (1−τ^{1−σ})/(1+τ^{1−σ}) (t−1) − [4 t μ̃ L τ^{1−σ}]/[μ(τ^{1−σ}+1)^2] · 1/[ m + (σ−1) (1−m+2(2−m))/((2−m)(1−m)) ] dm/dL_s. (eq. (43) as presented)
- Interpretation:
  - First two terms in (43) are negative and independent of matching complementarities; they work against agglomeration.
  - Third term S(τ) captures complementarities from matching (through dm/dLs and multipliers); dm/dLs is strictly negative and independent of τ, so S(τ) ≥ 0. S'(τ) < 0 and lim_{τ→∞} S(τ) = 0. S(τ) is maximal at τ = 1 (free trade).

### Break point: existence and numerical benchmark
- Definition: τ(B) is the critical τ > 1 satisfying dW_s/dL_s = 0. Symmetric equilibrium is robust if τ ≥ τ(B) and breaks when τ < τ(B).
- Numerical procedure: start at high trade costs with symmetric equilibrium, reduce τ in small steps to locate τ(B).
- Benchmark numerical result:
  - τ(B) = 1.19 (i.e., when trade costs fall below 19% of the producer price, firms could break symmetry by relocating from foreign to home).
- Comparative statics for τ(B):
  - Lower heterogeneity H reduces τ(B).
  - Lower training cost t reduces τ(B); elasticity of τ(B) to t is small.
  - Larger manufacturing share μ lowers τ(B): increasing μ from 0.4 to 0.6 reduces τ(B) from 1.19 to 1.16 in the benchmark.
  - Country-size asymmetry matters strongly: if one country is 20 per cent bigger than the other, break point in small country is close to free trade in virtually all cases, while break point in larger country increases substantially (to 1.31 in the benchmark).
  - There exists a robust equilibrium where agglomeration locates in the small country; at τ(B) ≤ 1.01 agglomeration can locate in the smaller home country (benchmark).
- Benchmark parameter values used in simulations:
  - H = 1, t = 1.1, L = L* = 100, σ = 4, α = 1/σ, β = (σ−1)/σ.
  - Initial N = N* = 40; deviation tested: one firm moving from foreign to home.

### Sustain point: robustness of agglomeration and numerical result
- Definition:
  - W*_s(τ) = maximum (zero-profit) wage rate a single firm in the foreign country can offer.
  - Agglomeration is sustainable at τ if W*_s(τ) < t.
  - Sustain point τ(S) defined by W*_s(τ(S)) = t.
- Method:
  - Consider all firms in home country; allow a single firm to establish in foreign country; solve profit-maximizing price and employment; check zero-profit wage against t/(1−H/2).
- Key demand/pricing expressions (preserved as in source):
  - Foreign firm demand:
    x* = p*^(−σ) μ L N p^(1−σ) Z(τ), where Z(τ) = τ^(1−σ)[2 (t−1) ˜μ + 1] + τ^(σ−1). (Equation (44))
  - Profit-maximizing price:
    p* = β σ/(σ−1) w*. (Equation (45))
  - Zero-profit wage per efficiency unit w* given by (46) (preserved as presented).
- Properties of Z(τ):
  - Z′(τ) = τ^(2(σ−1)−2) − 2 (t−1) μ/((1−μ)t+μ) − 1. (Equation (47))
  - Z(τ) has a single minimum at some τ > 1 and becomes positive and increasing for larger τ; Z(1) = 2 [(t−1) ˜μ + 1] = 2 t/μ (A-13).
- Existence and numerical sustain-point result:
  - As τ → ∞, Z(τ) → ∞ so w* increases and the workers’ willingness condition is eventually satisfied.
  - Numerical benchmark: τ(S) = 2.04 for the critical trade cost that induces relocation of 5% of all firms from home to foreign.
  - Interpretation: when trade costs exceed 2.04 the agglomeration equilibrium is not sustainable; symmetric equilibrium robustness persists at lower trade costs.
  - General pattern: reducing t or H reduces τ(S); τ(S) is generally more responsive to parameter changes than τ(B).
  - Typically τ(S) > τ(B); in the benchmark both equilibria are robust for τ between 1.19 and 2.04.
  - Group deviations: larger coordinated deviations reduce the critical trade-cost level for inducing relocation relative to single-firm deviations.

### Numerical examples (selected entries reported from the source table)
- Representative table entries (parameters: σ = 4, α = 1/σ, β = (σ−1)/σ; L = L* = 100 unless noted):
  - H = 1.5, t = 1.1, L = L* = 100, μ = 0.4 (both countries): τ(B) = 1.15, τ(S) = 1.25.
  - Benchmark row (H = 1.0, t = 1.1, L = L* = 100, μ = 0.4 both countries): τ(B) = 1.10, τ(S) = 1.19 (table entries preserved as in source).
  - Asymmetric example (H = 1.5, t = 1.1, L = 100 < L* = 120, μ = 0.4): Home: τ(B) = 3.05, τ(S) = 1.21; Foreign: τ(B) = 2.26, τ(S) = 1.01.
- Table interpretation:
  - Below τ(B) the symmetric equilibrium can be broken; above τ(S) the agglomeration equilibrium becomes unsustainable.

### Appendix: analytical derivations (break point and sustain point)
- Break point:
  - Differentiation leads to expression (A-3) and dm expressed in terms of dL_s (A-4)–(A-5).
  - Collecting terms yields coefficient on dW_s positive given μ < 1, τ ≥ 1, σ > 1; sign of dW_s/dL_s depends on coefficient on dL_s (eq. (43)).
  - S(τ) decomposed as A(τ) B(m) (A-7, A-8); at τ = 1, A(1) = t/μ and B > 1 for feasible m, yielding dW_s/dL_s > 0 (agglomeration favored at free trade).
- Sustain point:
  - Revenue p* x* uses Z(τ) (A-10); profit maximization and zero-profit condition lead to w* in (46) (A-12).
  - At τ = 1, Z(1) = 2 t/μ (A-13) and zero-profit wage evaluated in (A-14)–(A-17).
  - Free-trade agglomeration sustainability determined by inequality (A-18); there exists unique ˜m such that agglomeration is sustainable at zero trade costs for m < ˜m and unsustainable for m ≥ ˜m.
  - Simulations indicate ˜m is well above equilibrium values in typical parameterizations.

### Conclusions and testable implications
- Heterogeneous skills and imperfect matching create a force for agglomeration: firms prefer larger worker pools to improve matching despite monopoly power from product differentiation.
- Key regime results:
  - Agglomeration occurs when trade costs are sufficiently low (τ < τ(B)).
  - Once established, agglomeration can be sustained for a wide range of trade costs (up to τ(S)), and typically τ(S) > τ(B).
  - Larger economies are more likely to agglomerate, but agglomeration can locate in smaller countries if matching benefits dominate market-size advantages.
- Testable empirical implications:
  - More agglomeration in small high-tech industries than in larger/heavier industries.
  - Agglomeration should increase as trade costs fall and as task complexity increases.
- Suggested next step from source: empirical testing of these implications to assess whether observed agglomeration patterns are driven by skill matching.

*Source: _wp04103 - References and associated sections (IMF working paper).*

### References .............................................................................................................

### References ..........................................................................................................................25

### I. INTRODUCTION — main argument
- Demonstrates that when labor is heterogenous and the matching of skills with jobs is below first-best, the introduction of trade may lead to industrial agglomeration and interregional trade.
- Mechanism:
  - Agglomeration occurs because the average quality of matches improves when firms in the local market have a bigger pool of workers to choose from.
  - The force against agglomeration is the existence of trade costs.
  - At zero trade costs, regions that can trade always specialize. When there are positive trade costs, regions may or may not specialize depending on parameter values.
- Model features:
  - Heterogeneities in the performance of tasks by apparently similar workers: ex ante workers appear identical, but some are more productive in some jobs and others in other jobs.
  - Labor productivity depends on technology, training, conventional production factors, and the quality of the match between job and worker.
  - Firms with specialized skill requirements can recruit better-matched workers if they recruit in larger markets.
  - Combines with features from new trade theory: increasing returns, differentiated goods, and transport-costs.
- Empirical and anecdotal support (quotes preserved):
  - Sony UK general manager: “What keeps us here is the quality of the staff and the research and development capacity” (Financial Times, January 19, 2002).
  - Gavin Clarkson (software company owner, Choktaw Nation of Oklahoma): planning a technical training center to attract companies because, “Having a critical mass of people who are highly skilled and resourced is what attracts business to any location or community.” (Australian Financial Review,11 April 1999).
  - Financial Times: “The survival of struggling volume producers may or may not prove directly vital to the economy as a whole, but theirrole in providing skilled staff and components infrastructure for higher-margin niche manufacturers is hard to ignore.”
- Related prior focus: Krugman and Venables (1995) emphasized “components infrastructure”; this paper focuses on the role of “skilled staff.”

### Empirical evidence on labor pooling and matching
- Dumais, Ellison and Glaeser (2002) — Longitudinal Research Database (LRD) manufacturing data for the United States:
  - Examined Marshall’s three reasons for agglomeration: proximity to suppliers and customers, labor pooling, and information spillovers.
  - Found labor pooling was by far the most important force for agglomeration at the metropolitan area level.
  - New entrants tended to locate in areas where existing firms had labor requirements similar to their own.
- Interpretation and extensions:
  - Dumais, Ellison and Glaeser (2002) support labor pooling as an agglomeration force but do not distinguish reasons why labor pooling matters.
  - Indirect evidence supports match-differentiation being more important for advanced skills: routine tasks in less advanced economies afford less scope for differentiation; more complicated tasks allow varied types of performance.
  - Prediction: agglomeration should increase with economic growth and be more prevalent in industries requiring more high tech labor.
  - Dumais, Ellison and Glaeser (2002) find labor pooling especially strong in high technology industries.2
  - Footnote 2 (as in source): The main industries they list are fabricated metals, industrial machinery, electronic and electrical equipment, and instruments. Knowledge spillovers are also relatively more important for these industries but not as important as labor pooling. In accordance with our argument, Amiti and Cameron (2004) found that benefits of labor pooling were significant in a developing country (Indonesia), but were not as high as other agglomeration forces, such as interfirm linkages.
- Additional supporting evidence:
  - Dumais, Ellison and Glaeser (1997) find labor pooling appears more important in industries with more volatile employment.3
  - Footnote 3 (as in source): Rotemberg and Saloner (2000) examine skilled labor training and a hold-up problem; Krugman (1991) formalizes labor pooling via law of large numbers for idiosyncratic shocks.

### Relation to existing literature — three strands
- New economic geography:
  - Krugman (1991) and Krugman and Venables (1995) show agglomeration arises with interregional labor mobility or vertical input/output linkages.
  - Those models assume perfectly competitive labor markets and do not share the matching-based agglomeration reason here.
  - This paper does not include Krugman-style interregional labor mobility or vertical linkage agglomeration forces.
- Labor pooling literature:
  - Krugman (1991): labor pooling as insurance against idiosyncratic risk via law of large numbers when many firms are present.
  - Rotemberg and Saloner (2000): training costs and a hold-up problem; workers more likely to invest in training if many firms will compete for them.
  - Current paper’s labor-pooling mechanism differs from both Krugman’s and Rotemberg–Saloner’s mechanisms.
- External economies / matching externalities:
  - Henderson (1988) and Helsley and Strange (1990) discuss external economies and matching externalities in city-size models.
  - Helsley and Strange (1990): matching externalities push agglomeration, but land scarcity limits total agglomeration.
  - Contrast to current model:
    - Land is a free commodity in this paper.
    - Workers are of two types.
    - Location decisions are made by differentiated firms that can move between regions.

### Model structure and roadmap
- Section II:
  - Describes the model and derives choices of firms and workers.
  - Subsection II.A introduces the formal definition of heterogeneity and connects it to labor skills; derives the supply of labor to firms.
- Section III:
  - Derives the labor market equilibrium and (text cut off in source).

*Source: _wp04103 - References .............................................................................................................*

### section IV the aggregate equilibrium. It shows that there are at least two equilibria, a symmetric

### _wp04103 - section IV the aggregate equilibrium. It shows that there are at least two equilibria, a symmetric one with identical distribution offirms in each region and an agglomeration one with specialization in production and trade.

### The model — structure and assumptions
- Two regions (home and abroad), two sectors (agriculture and manufacturing), two skill levels (skilled and unskilled).
- Key mobility assumptions:
  - (a) perfect inter-sectoral and inter-regional mobility of firms;
  - (b) perfect inter-sectoral mobility of labor, but movement from agriculture to manufacturing requires a fixed “training” cost;
  - (c) no inter-regional mobility of labor.
- Production and preferences:
  - Agriculture: single good, linear technology, one worker produces one unit, price and agricultural wage normalized to 1.
  - Manufacturing: many differentiated varieties, one firm per variety, Dixit-Stiglitz preferences over manufacturing composite Cx and agricultural consumption Ca. Utility: Uk = vk Cxk^μ Cak^(1−μ), 0<μ<1, vk>0 (eq. (1)).
  - Manufacturing price index: Px defined in (4); iceberg transport costs τ, elasticity of substitution σ>1, τ≥1 (eq. (2) and surrounding text).
- Firms choose region and sector, then wage rate and output price; workers decide whether to train (become skilled) paying utility cost parameterized by t>1 and then consume based on realized wage.

### Labor market: matching, supply of effective labor, and wages
- Skill heterogeneity modeled on a circle of circumference 2H; uniform distribution of skilled workers around the circle; heterogeneity parameter H (H=0 implies no heterogeneity).
- Match quality measured by distance on circle; effective units of labor from a worker located at distance d from firm i is linear in distance, normalized to 1 at zero distance and expressed as 1−d (or zero if too far, but parameters ensure positive inputs).
- Firms locate symmetrically on the circle; with N manufacturing firms the distance between neighboring firms is 2H/N; worst mismatch H/N denoted m (mismatch).
- Wage posting and labor supply to a firm:
  - A firm posting wage wi atop market wage w attracts workers up to distance di given by di = (wi − w(1−2m)) / (wi + w) (eq. (14)).
  - Total effective labor units supplied to firm i: LEsi = (Ls/2H) (wi − w + 2mw)(wi + 3w − 2mw)/(wi + w)^2 (eq. (15)).
  - Elasticity of labor supply in symmetric equilibrium: ηEsi = (1−m)^2 (2−m) m^{-1}? (The source gives ηEsi ≡ (∂LEsi/∂wi)(wi/LEsi)|wi=w = (1−m)^2 (2−m) m / >0; see eq. (16).) [Preserve exact presentation: ηEsi = (1−m)^2 (2−m)m >0 as in (16).]
  - In symmetric equilibrium supply to each firm: LEsi = (Ls/N)(1−m/2) (eq. (17)).
- Occupational choice and training:
  - Training is a proportional utility cost via vk = 1/t if trained, vk = 1 otherwise; t>1.
  - Expected utility of a trained worker: Ūk = t^{−1} μ^{μ} (1−μ)^{1−μ} P_x^{−μ} P_a^{−(1−μ)} w(1−m/2) (eq. (19)).
  - Inter-sectoral mobility condition with both sectors active: w(1−m/2) = t > 1 (eq. (20)).
  - Feasible range for mismatch given w(1−m)≥1 and (20): m ≤ (2t−2)/(2t−1) (eq. (21)).

### Labor-market equilibrium — firm behavior, output, and number of firms
- Mark-up equation with symmetric supply elasticity substituted: p_i = w (1−m)^2/??? (source gives p_i = w (1−m)^2 σβ/(σ−1) but presented as eq. (22): p_i = w (1−m)^2 σβ/(σ−1).) (eq. (22)).
- Zero-profit condition and production lead to firm output:
  - x = α(σ−1)/β (1−m)^2 σ − (σ−1)(1−m)^2 (eq. (23)) [preserve exact expression as printed].
  - Demand for effective labor per firm: L_EDsi(m) = ασ / (σ − (σ−1)(1−m)^2 ) (eq. (24)).
- Supply to firm in terms of m: L_ESsi(m) = (Ls/H) m(1−m/2) (eq. (25)).
- Existence and uniqueness condition for interior equilibrium: solve L_ESsi(m) = L_EDsi(m), which after transformation M ≡ m(1−m/2) yields quadratic 2(σ−1) M^2 + M − (ασH/Ls) = 0 (eq. (27)). The single positive root with feasible m < 1 is the unique equilibrium m.
- Special case H = 0 (m = 0) yields N = Ls/(ασ) (eq. (26)).
- Comparative statics noted: equilibrium m increases in H and decreases in Ls; when more skilled workers enter, more manufacturing firms enter (complementarity).

### Aggregate equilibrium — symmetric and agglomeration allocations
- Define mean manufacturing wage Ws = t by using inter-sectoral mobility condition (re-writing eq. (20)) (eq. (28)).
- Implicit relation m = m(Ls), with m'(.) < 0 (eq. (29)).
- Equilibrium pricing (implicit in Ls): p = Ws/(1−m/2) (1−m)^2 σβ/(σ−1) (eq. (30)).
- National income with zero profits: Y = Ws Ls + Wa La, with Wa = 1 and La = L − Ls so Y = L + (Ws − 1) Ls (eqs. (31)–(32)).
- Demand for manufacturing output with trade (home and foreign) given respectively by eqs. (33) and (34).
- Profit-zero condition yields equation (35) for home country (and symmetric eq. (37) for foreign), introducing φ defined in (36):
  - φ ≡ (m/m^*) (μ p/p^*)^{σ−1} (eq. (36)).
- Symmetric equilibrium (μ < 0.5, equal region size, L = L^*): Ls = μ/[t − (t−1) μ] L ≡ μ̃ L (eq. (38)); then Ls^* = Ls, W_s^* = t, φ = 1.
- Equilibrium variables in symmetric case:
  - wage per unit w = t/(1−m/2) (from Ws = t),
  - firm price p as in eq. (39): p = t σ β (1−m)^2 / (1−m/2)^{(σ−1)} (eq. (39)),
  - output per firm x as in eq. (40): x = μ̃ L t^2 / [σ β m H (σ−1) (1−m)^2 (1−m/2)] (eq. (40)).
- Autarky limit: τ → ∞ yields no trade and the same factor-allocation equations apply.
- Agglomeration equilibrium (all manufacturing firms in home country: N^* = L_s^* = 0):
  - Home manufacturing employment Ls = μ(L + L^*) / [(1−μ) t + μ] = 2 μ̃ L (eq. (41)).
  - In agglomeration equilibrium foreign country produces only agricultural goods; home country has fraction 2 μ̃ of workers in manufacturing earning wage t.
  - All other equilibrium formulae (wage per unit, price) remain given by the same expressions but firm output per firm is twice the symmetric-equilibrium expression in (40).

### Selection of equilibrium — robustness, break point, and sustain point
- With trade, both symmetric and agglomeration equilibria exist. Selection is analyzed by a robustness (stability) criterion via optimal deviations (following Fujita et al., 1999 style).
- Method: start from symmetric equilibrium and perturb by moving manufacturing workers dLs from foreign to home (dL^*_s = −dLs). Evaluate sign of dWs/dLs:
  - If dWs/dLs < 0, symmetric equilibrium is robust.
  - If dWs/dLs > 0, symmetric equilibrium breaks (a “break point” in trade costs exists where deviation is profitable and the symmetric equilibrium is not robust).
- Total differentiation of (35) at symmetric equilibrium yields (42):
  - μ̃ L (μ) dW_s + t μ dL_s = (1 − τ^{1−σ})/(1 + τ^{1−σ}) [(t−1) dL_s + μ̃ L dW_s] − [2 τ^{1−σ} t μ̃ L / μ (1 + τ^{1−σ})^2] dφ (eq. (42)).
- The sign of dW_s/dL_s is the same as the sign of expression (43) (reproduced exactly):
  - − t/μ + (1−τ^{1−σ})/(1+τ^{1−σ}) (t−1) − [4 t μ̃ L τ^{1−σ}]/[μ(τ^{1−σ}+1)^2] · 1/[ m + (σ−1) (1−m+2(2−m))/((2−m)(1−m)) ] dm/dL_s. (eq. (43) as presented)
- Interpretation of terms in (43):
  - First two terms are negative and independent of matching complementarities; they work against agglomeration.
  - The third term (denote S(τ)) captures the complementarities from matching (through dm/dLs and other multipliers); its magnitude and sign determine whether complementarities can overturn the negative first two terms.
  - Appendix results: dm/dLs is strictly negative and independent of τ, so S(τ) ≥ 0. S'(τ) < 0 and lim_{τ→∞} S(τ) = 0. Therefore, at very high trade costs symmetry is not broken. The maximum S(τ) is reached at τ = 1 (free trade).

*Italic: Content summarized from _wp04103 - section IV the aggregate equilibrium. It shows that there are at least two equilibria, a symmetric one with identical distribution offirms in each region and an agglomeration one with specialization in production and trade.*

### Appendix shows thatS(1)is sufficiently large to make the entire expression in (43) positive, so in

### _wp04103 - Appendix shows thatS(1)is sufficiently large to make the entire expression in (43) positive, so in

### Break point and existence of agglomeration
- There is a critical value of τ>1, denoted τ(B), that satisfies dW_s/dL_s = 0; the symmetric equilibrium is robust to deviations if τ ≥ τ(B) and breaks when τ < τ(B).
- Numerical procedure used:
  - Start from an initial condition with sufficiently high trade costs and a symmetric equilibrium.
  - Reduce trade costs in small steps and locate the critical τ that switches the equilibrium from symmetry to agglomeration.
- Benchmark result:
  - τ(B) = 1.19 (i.e., when trade costs fall below 19% of the producer price, firms could break the symmetric equilibrium by relocating from the foreign country to the home country).
- Comparative statics for τ(B):
  - Lower heterogeneity in labor skills (lower H) reduces τ(B).
  - Lower training costs (lower t) reduce τ(B); the elasticity of τ(B) to training costs is small.
  - A larger manufacturing sector (increase in μ) lowers τ(B): increasing μ from 0.4 to 0.6 reduces τ(B) from 1.19 to 1.16 in the benchmark.
  - Country size matters strongly: if one country is 20 per cent bigger than the other, the break point in the small country is close to free trade in virtually all cases, while the break point in the larger country increases substantially (to 1.31 in the benchmark case).
  - There exists a robust equilibrium where agglomeration locates in the small country; at τ(B) ≤ 1.01 the agglomeration can locate in the smaller home country (benchmark).

- Benchmark parameter values for simulations:
  - H = 1, t = 1.1, L = L* = 100, σ = 4, α = 1/σ, β = (σ−1)/σ.
  - Initial number of firms is set at N = N* = 40 and the deviation tested is whether one firm can profitably move from foreign to home.

### Sustain point and robustness of agglomeration
- Definition:
  - Let W*_s(τ) be the maximum (zero-profit) wage rate that a single firm in the foreign country can offer. The agglomeration equilibrium is sustainable at τ if W*_s(τ) < t.
  - The sustain point τ(S) is defined by W*_s(τ(S)) = t.
- Mechanism:
  - Consider all firms located in the home country; allow a single manufacturing firm to establish in the foreign country. Solve for the firm’s profit-maximizing price and employment, and check whether the zero-profit wage offered exceeds t/(1−H/2) (workers’ willingness condition).
- Demand and pricing used:
  - Foreign firm demand: x* = p*^(−σ) μ L N p^(1−σ) Z(τ), where Z(τ) = τ^(1−σ)[2 (t−1) ˜μ + 1] + τ^(σ−1). (Equation (44))
  - Profit-maximizing price: p* = β σ/(σ−1) w*. (Equation (45))
  - Zero-profit wage per efficiency unit: w* given by (46):
    w* = (σ−1)/(βσ) [ μ/(α(σ−1)/β · N p^(1−σ) μ L Z(τ)) ]^(−1/σ)
    (preserved as presented in the source).
- Key properties of Z(τ):
  - Z′(τ) = τ^(2(σ−1)−2) − 2 (t−1) μ/((1−μ)t+μ) − 1. (Equation (47))
  - At τ = 1, Z′(τ) is negative, but Z(τ) has a single minimum at some τ > 1 and becomes positive and increasing for larger τ.
- Sustain point existence:
  - As τ → ∞, Z(τ) → ∞, so w* increases indefinitely and (48) (workers’ willingness condition) is eventually satisfied.
  - Need to show that at τ = 1, (48) is not satisfied for small feasible values of m (the Appendix shows this is the case), establishing a range of trade costs including free trade that guarantee robustness of the agglomeration equilibrium.
  - There may be high values of m for which the agglomeration equilibrium is not robust even at zero trade costs: when m is large, a firm may prefer to locate in the foreign country to avoid local competition despite worse matching.
  - If manufacturing is very small, there may not be sufficient benefits from agglomeration to make it sustainable.
- Numerical sustain-point result:
  - In the benchmark case, the critical trade cost that induces relocation of 5% of all firms from the home country to the foreign country is τ(S) = 2.04.
  - Interpretation: when trade costs exceed 2.04, the agglomeration equilibrium is not sustainable; symmetric equilibrium robustness persists at even lower trade costs.
  - General pattern: reducing either t or H reduces the sustain point; τ(S) is generally more responsive to parameter changes than τ(B).
  - In general τ(S) > τ(B), so there is an interval (τ(B), τ(S)) where both the symmetric and agglomeration equilibria are robust to small deviations; in the benchmark both equilibria are robust for τ between 1.19 and 2.04.
- Group deviations:
  - If larger groups of firms deviate together, the critical trade-cost level for inducing relocation is lower than for a single firm or 5% deviation.

### Table 1 (selected numerical values and interpretation)
- Table displays break and sustain points for trade costs at different parameter values (L = L* = 100; L = 100 < L* = 120; μ = 0.4 and μ = 0.6; home vs. foreign asymmetries).
- Representative entries (each row corresponds to given H and t; columns provide τ(B) and τ(S) for both countries or for home/foreign when asymmetric):
  - For H = 1.5, t = 1.1, L = L* = 100, μ = 0.4 (both countries): τ(B) = 1.15, τ(S) = 1.25.
  - For H = 1.5, t = 1.1, L = 100 < L* = 120, μ = 0.4 (Home): τ(B) = 3.05, τ(S) = 1.21; (Foreign): τ(B) = 2.26, τ(S) = 1.01; and other asymmetric entries as given in Table 1.
  - For the benchmark row (H = 1.0, t = 1.1, L = L* = 100, μ = 0.4 both countries): τ(B) = 1.10, τ(S) = 1.19 (table entries preserved exactly as in source).
- Notes accompanying Table 1:
  - Other parameter values: σ = 4, α = 1/σ, β = (σ−1)/σ.
  - H measures the degree of heterogeneity and t the training cost.
  - At trade costs below τ(B) the symmetric equilibrium can be broken and at points above τ(S) the agglomeration equilibrium becomes unsustainable.

### Appendix — analytical derivations (A. Break point; B. Sustain point)
- Break point derivation highlights:
  - Differentiation of key equilibrium expressions yields dφ in terms of dm, dp, and dW_s; substitution and imposition of symmetry produces expression (A-3).
  - Total differentiation of m from (27) gives dm in terms of dL_s (A-4) and reorganized in (A-5).
  - Collecting terms yields coefficient on dW_s shown in (A-6); this coefficient is positive given μ < 1, τ ≥ 1, σ > 1. Thus sign of dW_s/dL_s depends on the coefficient on dL_s (equation (43)).
  - S(τ) can be written as product A(τ) B(m) (A-7 and A-8). In free trade A(1) = t/μ and the sign of the whole expression reduces to sign of t/μ (B−1) (A-9). The fractions in B exceed 1 for feasible m so B > 1 and dW_s/dL_s > 0.
- Sustain point derivation highlights:
  - Firm revenue p* x* uses Z(τ) from (A-10).
  - Profit maximization with production L* E_s = α + β x* yields price p* in (45) and zero-profit condition (A-12); solving gives w* in (46).
  - At τ = 1, Z(1) = 2 [(t−1) ˜μ + 1] = 2 t/μ (A-13).
  - The zero-profit wage at τ = 1 is given in (A-14) and transformed using equilibrium conditions to (A-15)–(A-17).
  - Free trade equilibrium is not sustainable if inequality (A-18) holds; analysis shows there exists a unique ˜m such that agglomeration is sustainable at zero trade costs for m < ˜m and unsustainable for m ≥ ˜m.
  - The Appendix notes it is not possible to determine analytically whether ˜m is above or below equilibrium m, but simulations indicate ˜m is well above equilibrium values; ˜m tends to be a fairly large number within the feasible range.

### Conclusions (summary of substantive implications)
- Heterogeneity of skills can induce agglomeration of economic activity even when it gives firms monopoly power; firms prefer markets with larger pools of workers to improve matching.
- Agglomeration occurs when trade costs are sufficiently low (τ < τ(B)) and, once established, can be sustained over a wide range of trade costs (up to τ(S)).
- Larger economies are more likely to exhibit agglomeration; however, agglomeration can also locate in smaller countries when matching benefits in the skilled labor market outweigh market-size benefits.
- Testable implications for empirical research:
  - More agglomeration in small high-tech industries than in larger/heavier industries.
  - Agglomeration should increase as trade costs fall and as task complexity increases.
- Suggested next step: empirical testing of these implications to assess whether observed agglomeration patterns are driven by skill matching.

*Source: IMF working paper content (Appendix and associated sections).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2004/_wp04103.pdf_